Method for interference alignment in wireless network

ABSTRACT

A method for interference alignment in wireless network having 3 transmitters and 3 receivers which are equipped with M antennas is provided. The method comprising: transmitting, performed by each of the 3 transmitters, a pilot signal known to the 3 receivers; estimating, performed by each of the 3 receivers, each channel from transmitter; transmitting, performed by each of the 3 receivers, feedback information to target transmitter; and determining, performed by transmitter 2 and transmitter 3, a precoding vector; wherein a degree of freedom (DoF) of a transmitter 1 is (M/2−α), a DoF of the transmitter 2 or the transmitter 3 is M/2.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is the National Stage filing under 35 U.S.C. 371 ofInternational Application No. PCT/KR2011/000753, filed on Feb. 1, 2011,which claims the benefit of U.S. Provisional Application Ser. Nos.61/417,285, filed on Nov. 26, 2010, 61/364,395, filed on Jul. 15, 2010,and 61/300,791, filed on Feb. 2, 2010, the contents of which are allincorporated by reference herein in their entirety.

TECHNICAL FIELD

The present invention relates to wireless communication, and moreparticularly, to a method for interference alignment in wirelessnetwork.

BACKGROUND ART

In a wireless network with multiple interfering links, interferencealignment (IA) is used. IA is a transmission scheme achieving linear sumcapacity scaling with the number of data links, at high SNR. With IA,each transmitter designs the precoder to align the interference on thesubspace of allowable interference dimension over the time, frequency orspace dimension, where the dimension of interference at each receiver issmaller than the total number of interferers. Therefore, each receiversimply cancels interferers and acquires interference-free desired signalspace using zero-forcing (ZF) receive filter.

Most of conventional research on IA is considered to achieve the maximumgain of IA using the infinite selectivity over symbol extensions, whichis unrealistic in practical wireless networks. Therefore, the recentstudies on MIMO IA focused on the design of IA precoder using a finitespace dimension over one transmission slot, which is called MIMOconstant channel.

The difficulties of IA in MIMO constant channel is to derive the closedform solution of the IA precoder. In other word, conventionalinterference alignment (IA) solution for achieving the optimal degreesof freedom (DoF) requires product of all cross link channel informationsince all precoders are coupled. These coupled condition requireschannel matrix multiplication. If channel state information attransmitter is imperfect, inaccurate channel matrix multiplicationarises error amplification due to summation and multiplication of error.Therefore, IA solution for achieving the optimal DoF may not be optimalin practical system. To avoid the product of channel matrices to getbetter performance and reduce feedback overhead, efficient IA method isneeded.

SUMMARY OF INVENTION Technical Problem

It is an object of the present invention to provide a method ofinterference alignment in wireless network.

Solution to Problem

A method for interference alignment in wireless network having 3transmitters and 3 receivers which are equipped with M antennas isprovided. The method comprising: transmitting, performed by each of the3 transmitters, a pilot signal known to the 3 receivers; estimating,performed by each of the 3 receivers, each channel from transmitter;transmitting, performed by each of the 3 receivers, feedback informationto target transmitter; and determining, performed by transmitter 2 andtransmitter 3, a precoding vector; wherein a degree of freedom (DoF) ofa transmitter 1 is (M/2−α), a DoF of the transmitter 2 or thetransmitter 3 is M/2.

Advantageous Effects of Invention

In accordance with the present invention, it is provided more errorrobust than conventional IA method and a reduction of feedback overhead.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows a framework of the interference alignment.

FIG. 2 shows a CSI exchange method (method 1).

FIG. 3 illustrates a conventional full-feedback method.

FIG. 4 shows an example of star feedback method (method 3).

FIG. 5 shows the procedure of CSI exchange method in case that K is 4.

FIG. 6 shows the procedure of CSI exchange method based on precoded RSin case that K is 4.

FIG. 7 illustrates feedback information at each receiver for efficientinterference alignment method for 3 cell MIMO interference channel withlimited feedback.

FIG. 8 shows sum rate performance, total number of stream of the methodof FIG. 7 and the other methods.

MODE FOR THE INVENTION

In a wireless network with multiple interfering links, interferencealignment (IA) uses precoding to align at each receiver the interferencecomponents from different sources. As a result, in the high SNR regime,the network capacity scales logarithmically with thesignal-to-noise-ratio (SNR) and linearly with half of the number ofparallel sub-channels, called degrees of freedom (DoF). In addition, inthe presence of multi-antennas, the capacity also scales linearly withthe spatial DoF per link. IA has been proved to be optimal in terms ofDoF.

This motivates extensive research on IA methods for various types ofchannels and settings, including MIMO channel, cellular networks,distributed IA, IA in the signal space and limited feedback. However,most existing works on IA rely on the impractical assumption that eachtransmitter in an IA network requires perfect CSI of all interferencechannels. Some preliminary results have been obtained on the scalinglaws of numbers of CSI feedback bits with respect to the SNR under theIA constraint. However, there exist no designs of practical CSI feedbackalgorithms for IA networks.

To facilitate the description of present invention, the framework of theInterference alignment related to the present invention will beexplained firstly.

FIG. 1 shows a framework of the interference alignment.

Referring FIG. 1, the framework of the the interference alignmentcomprises a cooperative CSI exchange, a transmission power control and ainterference misalignment control.

1. Cooperative CSI exchange: Transmit beamformers will be alignedprogressively by iterative cooperative CSI exchange between interferersand their interfered receivers. In each round of exchange, a subset oftransmitters receive CSI from a subset of receivers and relay the CSI toa different subset of receivers. Based on this approach, CSI exchangemethods are optimized for minimizing the number of CSI transmissionlinks and the dimensionality of exchanged CSI, resulting in smallnetwork overhead.

2. Transmission power control: Given finite-rate CSI exchange,quantization errors in CSI cause interference misalignment. To controlthe resultant residual interference, methods are invented to supportexchange of transmission power control (TPC) signals between interferersand receivers to regulate the transmission power of interferers underdifferent users' quality-of-service (QoS) requirements. Since IA isdecentralized and links are coupled, multiple rounds of TPC exchange maybe required.

3. Interference misalignment control: Besides interferers' transmissionpower, another factor that influences interference power is the degreesof interference misalignment (IM) of transmit beamformers, whichincrease with CSI quantization errors and vice versa. The IM degrees areregulated by adapting the resolutions of feedback CSI to satisfy users'QoS requirements. This requires the employment of hierarchical codebooksat receivers that support variable quantization resolutions. DesigningIM control policies is formulated as an optimization problem ofminimizing network feedback overhead under the link QoS constraints.Furthermore, the policies are optimized using stochastic optimization tocompress feedback in time.

In addition, it will be described a star feedback method, a efficientinterference alignment for 3-cell MIMO interference channel with limitedfeedback.

4. Star Feedback Method: For achieving IA, each precoder is aligned toother precoders and thus its computation can be centralized at centralunit, which is called CSI base station (CSI-BS). In star feedbackmethod, CSI-BS gathers CSI of interference from all receivers andcomputes the precoders. Then, each receiver only feeds CSI back toCSI-BS, not to all transmitters. For this reason, star feedback methodcan effectively scale down CSI overhead compared with conventionalfeedback method.

5. Efficient interference alignment for 3 cell MIMO interference channelwith limited feedback: Conventional IA solution for achieving theoptimal DoF and a special case IA solution requires product of all crosslink channel information since all precoders are coupled. When limitedfeedback channel is assumed, channel matrix product arises a criticalproblem that there is a K-fold increase in error of initial precoder dueto multiplied and summed channel quantization errors. IA solution forachieving the optimal DoF may not be optimal in practical number offeedback bits.

To avoid the product of channel matrices and get better performance, aDoF following method is proposed. If one node does not use α DoF, IAsolution is separated to several independent equation, resulting inavoiding the product of all cross link channel matrices and reducingfeedback overhead.

Firstly, a system model is described.

I. System Model

we introduce the system model of K user MIMO interference channel anddefine the metric of network overhead that is required for implementingIA precoder.

Consider K user interference channel, where K transmitter-receiver linksexist on the same spectrum and each transmitter send an independent datastream to its corresponding receiver while it interferers with otherreceivers. Assuming every transmitter-receiver node is equipped with Mantennas, the channel between transmitter and receiver is modeled as M×Mindependent MIMO block channel consisting of path-loss and small-scalefading components. Specifically, it is denoted the channel from the j-thtransmitter and the k-th receiver as d_(kj) ^(−α/2) H^([kj]), where α isthe path-loss exponent, d_(kj) is the distance between transmitter andreceiver and H^([kj]) is M×M matrix of independently and identicallydistributed circularly symmetric complex Gaussian random variables withzero mean and unit variance, denoted as CN(0,1).

Let denote v^([k]) and r^([k]) as a (M×1) beamforming vector and receivefilter at the k-th transceiver, where ∥v^([k])∥²=∥r^([k])∥²=1. Abeamforming vector can be called other terminologies such as beamformeror a precoding vector. Then, the received signal at the k-th receivercan be expressed as below equation.

$\begin{matrix}{y^{\lbrack k\rbrack} = {{H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}s_{k}} + {\sum\limits_{j \neq k}{H^{\lbrack{kj}\rbrack}v^{\lbrack j\rbrack}s_{j}}} + n_{k}}} & \left\lbrack {{equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

and the sum rate is calculated as below equation.

$\begin{matrix}{R_{sum} = {\sum\limits_{k = 1}^{K}{\log_{2}\left( {1 + \frac{\frac{P}{d_{kk}^{\alpha}}{{r^{{\lbrack k\rbrack}\dagger}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{{\sum\limits_{k \neq j}{\frac{P}{d_{kj}^{\alpha}}{{r^{{\lbrack k\rbrack}\dagger}H^{\lbrack{kj}\rbrack}v^{\lbrack j\rbrack}}}^{2}}} + \sigma^{2}}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

where s_(k) denotes a data symbol sent by the k-th transmitter withE[|s_(k|) ²]=P and n_(k) is additive white Gaussian noise (AWGN) vectorwith covariance matrix σ²I_(m).

Under the assumption of perfect and global CSI, IA aims to aligninterference on the lower dimensional subspace of the received signalspace so that each receiver simply cancels interferers and acquires Kinterference-free signal space using zero-forcing (ZF) receive filtersatisfying following equation.r ^([k]†) H ^([kj]) v ^([j])=0,∀k≠j  [equation 3]

Therefore, the achievable sum rate of IA is computed by below equation.

$\begin{matrix}{R_{sum} = {\sum\limits_{k = 1}^{K}{\log_{2}\left( {1 + {\frac{P}{\sigma^{2}}d_{kk}^{- \alpha}{{r^{{\lbrack k\rbrack}\dagger}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

It is assumed that each receiver, say the m-th receiver, perfectlyestimates all interference channels, namely the set of matrices{H^([mk])}_(k=1) ^(K). Also, it is considered the case where CSI can besent in both directions between a transmitter and a receiver.

II. CSI Feedback Methods

In MIMO constant channel, the closed form solution of IA in K=3 user ispresented in prior arts. And the solution of K=M+1 with a single datastream at transmitter-receiver pairs is proposed in prior arts. However,such closed form solutions in general K user interference channel arestill open problem. Here, three CSI feedback methods, namely the 1. CSIexchange method (method 1), 2. modified CSI exchange method (method 2)and 3. star feedback method (method 3) are described for achieving IAunder the constraints K=M+1. Also, we compare the efficiency of eachfeedback method with sum overhead defined as the total number of complexCSI coefficients transmitted in the network for a given channelrealization expressed in equation 5.

$\begin{matrix}{N = {\sum\limits_{m,{k \in {\{{1,2,\ldots,K}\}}}}\left( {N_{TR}^{\lfloor{mk}\rfloor} + N_{RT}^{\lfloor{mk}\rfloor}} \right)}} & \left\lbrack {{equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In equation 5, N_(TR) ^([mk]) denotes the integer equal to the number ofcomplex CSI coefficients sent from the k-th receiver to the m-thtransmitter, and N_(RT) ^([mk]) from the k-th transmitter to the m-threceiver.

FIG. 2 shows a CSI exchange method (method 1).

Assuming each pair of transmitter-receiver gets one multiplexing gainunder the constraint K=M+1, the set of interference{H^([mk])v^([k])}_(k=1; k≠m) ^(K) at each receiver should be aligned onthe subspace of dimension at most M−1 to satisfy IA condition whichdescribed in equation 3. That is to say, at least two of all interferersare designed on the same subspace at the receiver as below equation.

$\begin{matrix}{{{{span}\mspace{11mu}\left( {H^{\lbrack{{1K} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack{1K}\rbrack}v^{\lbrack K\rbrack}} \right)}}{{{span}\mspace{11mu}\left( {H^{\lbrack{2K}\rbrack}v^{\lbrack K\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack 21\rbrack}v^{\lbrack 1\rbrack}} \right)}}{{{span}\mspace{11mu}\left( {H^{\lbrack 31\rbrack}v^{\lbrack 1\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack 32\rbrack}v^{\lbrack 2\rbrack}} \right)}}\vdots{{{span}\mspace{11mu}\left( {H^{\lbrack{{KK} - 2}\rbrack}v^{\lbrack{K - 2}\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack{{KK} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

In equation 6, span(A) denotes the vector space that spanned by thecolumns of A and each equation satisfies that span(H^([k+2k])v^([k]))=span (H^([k+2k+1])v^([k+1])) (i.e.span(v^([k]))=span (H^([k+2k]))⁻¹H^([k+2k+1])v^([k+1]))). Asconcatenating {span (v^([k]))}_(k=1) ^(K), IA beamformers are computedby below equation and then normalized to have unit norm.

$\begin{matrix}{{v^{\lbrack 1\rbrack} = {{any}\mspace{14mu}{eigenvector}\mspace{14mu}{of}\mspace{14mu}\left( H^{\lbrack 21\rbrack} \right)^{- 1}{H^{\lbrack{2K}\rbrack}\left( H^{\lbrack{1K}\rbrack} \right)}^{- 1}H^{\lbrack{{1K} - 1}\rbrack}\mspace{14mu}\ldots\mspace{14mu}\left( H^{\lbrack 32\rbrack} \right)^{- 1}H^{\lbrack 31\rbrack}}}\mspace{79mu}{v^{\lbrack 2\rbrack} = {\left( H^{\lbrack 32\rbrack} \right)^{- 1}H^{\lbrack 31\rbrack}v^{\lbrack 1\rbrack}}}\mspace{79mu}\vdots\mspace{79mu}{v^{\lbrack{K - 1}\rbrack} = {\left( H^{\lbrack{{KK} - 1}\rbrack} \right)^{- 1}H^{\lbrack{{KK} - 2}\rbrack}v^{\lbrack{K - 2}\rbrack}}}\mspace{79mu}{v^{\lbrack K\rbrack} = {\left( H^{\lbrack{1K}\rbrack} \right)^{- 1}H^{\lbrack{{1K} - 1}\rbrack}v^{\lbrack{K - 1}\rbrack}}}} & \left\lbrack {{equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

The solution of k-th beamformer v^([k]) in equation 7 is seriallydetermined by the product of pre-determined v^([k−]) and channel matrix(H^([k+1k−1]))⁻¹H^([k+1k−1]).

Hereinafter, using the property that v^([k]) is serially determined, CSIexchange method (method 1) will be described. Hereinafter, it is assumedthat K=4, M=3.

1. First, transmitters determine which interferers are aligned on thesame subspace at each receiver. Then, each transmitter informs itscorresponding receiver of the indices of two interferers to be alignedin the same direction. This process is called set-up.

2. Receiver k(∀k≠3) feeds back the product matrix (H^([21]−1)H^([2K]), .. . , (H^([32]))⁻¹H^([31]) to receiver 3 which determines v^([1]) as anyeigenvector of (H^([21]))⁻¹H^([2K])(H^([1K]))⁻¹H^([1K−1]). . .(H^([32]))⁻¹H^([31]). Then, v^([1]) is fed back to transmitter 1.

3. Computation of v^([2]), . . . , v^([K−1]): Transmitter k−1 feedsforward v^([k−1]) to receiver k+1. Then, receiver k+1 calculates v^([k])using equation 7 and feeds back v^([k]) to transmitter k. This processis performed for k=2 to K−1.

4. computation of v^([k]).

Transmitter K−1 feeds forward v^([K−1]) to receiver 1. Then, receiver 1calculates v^([K]) using equation 7 and feeds back v^([K]) totransmitter K.

In CSI exchange method, two interferers are aligned at each receiveramong K−1 interferers. Therefore, the receiver requires the indices oftwo interferers to be aligned before starting the computation andexchange procedure of {v^([1]), . . . , v^([K])}. The signals for aset-up are forwarded through a control channel from transmitter tocorresponding receiver. These set-up signals are forwarded throughB_(setup) bits control channel from transmitter to correspondingreceiver and its overhead is computed in following lemma.

Lemma 1. The set-up overhead for the CSI exchange method is given asbelow equation.

$\begin{matrix}{B_{setup} = \left\lceil {K \cdot {\log_{2}\left( \frac{\left( {K - 1} \right)\left( {K - 2} \right)}{2} \right)}} \right\rceil} & \left\lbrack {{equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

Each receiver has K−1 interferers from other transmitters. Therefore,_(K−1)C₂ groups of aligned interferers exist at each receiver, where_(K−1)C₂=(K−1)(K−2)/2. To inform each of K receivers about the group ofaligned interferers, total

$\left\lceil {K \cdot {\log_{2}\left( \frac{\left( {K - 1} \right)\left( {K - 2} \right)}{2} \right)}} \right\rceil$

bits are required for the set-up signaling.

Once the aligned interferers are indicated at the receiver, eachreceiver feeds back CSI of the selected interferers in given channelrealization and the beamformers v^([1]), v^([2]), . . . , v^([K]) aresequentially determined by method 1. The overhead for the CSI exchangemethod can be measured in terms of the number of exchanged complexchannel coefficients. Such overhead is specified in the followingproposition.

Proposition 1

For the CSI exchange method in MIMO channel, the network overhead isgiven as below equation.N _(EX)=(K−1)M ²+(2K−1)M  [equation 9]

All receivers inform CSI of the product channel matrices to receiver 3,which comprises (K−1)M² nonzero coefficients. After computing thebeamformer v^([1]), M nonzero coefficients are required to feed it backto transmitter 1. Each beamformer is determined by iterative precoderexchange between transmitters and interfered receivers. In each round ofexchange, the number of nonzero coefficients of feedforward and feedbackbecomes 2M. Thus, total network overhead comprises(K−1)M ²+(2K−1)M.

FIG. 3 illustrates a conventional full-feedback method.

For comparison, the sum overhead for the convention CSI feedback methodis illustrated. In the existing IA literature, the design of feedbackmethod is not explicitly addressed. Existing works commonly assume CSIfeedback from each receiver to all its interferers, corresponding to thefull-feedback method illustrated in FIG. 3. For such a conventionalmethod, each receiver transmits the CSI{H[mk]}_(k=1) ^(K) to each of itsK−1 transmitters and the resultant sum overhead is given in thefollowing lemma.

Lemma 2. The sum overhead of the full-feedback method is given as belowequation.N _(FF) =K ²(K−1)M ²  [equation 10]

Each receiver feeds back K×M² nonzero coefficients to K−1 interferers.Since K receivers feed back CSI, total network overhead comprisesK²(K−1)M².

From the above result, the overhead N_(FF) increases approximately asK³M² whereas the network throughput grows linearly with K. Thus thenetwork overhead is potentially a limiting factor of the networkthroughput. By comparing proposition 1 and lemma 2, with respect to thefull-feedback method, the CSI exchange method requires much less sumoverhead for achieving IA, namely on the order of KM².

In the CSI exchange method described above, v^([1]) is solved by theeigenvalue problem that incorporates the channel matrices of allinterfering links. However, it still requires a huge overhead innetworks with many links or antennas. To suppress CSI overhead ofv^([1]), we may apply the follow additional constraints indicated by abelow equation.span(H ^([i1]) v ^([1]))=span(H ^([i2]) v ^([2]))span(H ^([i+11]) v ^([1]))=span(H ^([i+12]) v ^([2]))  [equation 11]

In equation 11, i is not 1 or 2. As v^([1]) and v^([2]) are aligned onsame dimension at receiver i and i+1, v^([1]) is computed as theeigenvalue problem of (H^([i1]))⁻¹H^([i2])(H^([i+12]))⁻¹H^([i+11]) thatconsists of two interfering product matrices.

Applying the properties expressed in equation 11 at receiver K−1 and K,it can be reformulated IA condition under K=M+1 constraints as belowequation.

$\begin{matrix}{{{{span}\mspace{11mu}\left( {H^{\lbrack 12\rbrack}v^{\lbrack 2\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack 13\rbrack}v^{\lbrack 3\rbrack}} \right)}}{{{span}\mspace{11mu}\left( {H^{\lbrack 23\rbrack}v^{\lbrack 3\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack 24\rbrack}v^{\lbrack 4\rbrack}} \right)}}\vdots{{{span}\mspace{11mu}\left( {H^{\lbrack{K - 11}\rbrack}v^{\lbrack 1\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack{K - 12}\rbrack}v^{\lbrack 2\rbrack}} \right)}}{{{span}\mspace{11mu}\left( {H^{\lbrack{K\; 1}\rbrack}v^{\lbrack 1\rbrack}} \right)} = {{span}\mspace{11mu}\left( {H^{\lbrack{K\; 2}\rbrack}v^{\lbrack 2\rbrack}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

It follows that v^([1]), v^([2]), . . . , v^([K]) can be selected asbelow equation.

$\begin{matrix}{{v^{\lbrack 1\rbrack} = {{eigenvector}\mspace{14mu}{of}\mspace{14mu}\left( H^{\lbrack{K - 11}\rbrack} \right)^{- 1}{H^{\lbrack{K - 12}\rbrack}\left( H^{\lbrack{K\; 2}\rbrack} \right)}^{- 1}H^{\lbrack{K\; 1}\rbrack}}}\mspace{79mu}{v^{\lbrack 2\rbrack} = {\left( H^{\lbrack{K\; 2}\rbrack} \right)^{- 1}H^{\lbrack{K\; 1}\rbrack}v^{\lbrack 1\rbrack}}}\mspace{79mu}{v^{\lbrack 3\rbrack} = {\left( H^{\lbrack 13\rbrack} \right)^{- 1}H^{\lbrack 12\rbrack}v^{\lbrack 2\rbrack}}}\mspace{85mu}\vdots\mspace{79mu}{v^{\lbrack K\rbrack} = {\left( H^{\lbrack{K - {2K}}\rbrack} \right)^{- 1}H^{\lbrack{K - {2K} - 1}\rbrack}{v^{\lbrack{K - 1}\rbrack}.}}}} & \left\lbrack {{equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

Using the equation 13, the CSI exchange method can be modified. Themodified CSI exchange method can be called a method 2 for a convenience.The method 2 comprises following procedures.

1. First, transmitters determine which interferers are aligned on thesame subspace at each receiver. Then, each transmitter informs itscorresponding receiver of the indices of two interferers to be alignedin the same direction (set-up).

2. Receiver K forwards the matrix (H^([K2]))⁻¹H^([K1]) to receiver K−1.Then, receiver K−1 computes v^([1]) as any eigenvector of(H^([K−11]))⁻¹H^([K−12])(H^([K2]))⁻¹H^([K1]) and feeds back it to thetransmitter 1.

3. Transmitter 1 feeds forward v^([1]) to receiver K. Then, receiver Kcalculates v^([2]) and feeds back v^([2]) to transmitter 2.

4. Computation of v^([3]), . . . , v^([K])

Transmitter k−1 feeds forward v^([k−1]) to receiver k−2. Then, receiverk−2 calculates v^([k]) and feeds back v^([k]) to transmitter k−1. Thisprocess is performed for k=3 to K.

The corresponding sum overhead of modified CSI exchange method (method2) is given in the following equation.N _(MEX) =M ²+(2K−1)M  [equation 14]

Comparing the method 2 with the method 1, both methods show the sameburden of overhead for the exchange of beamformers. However, the productchannel matrix for v^([1]) in Modified CSI exchange method requiresconstant M² overhead in any K user case while overhead of v^([1]) in CSIexchange method increases with KM².

The CSI exchange methods (method 1, method 2) in previous descriptiondegrade the amount of network overhead compared with conventional fullfeedback method. However, it requires 2K−1 iterations for the exchangeof beamformers in method 1 and 2. As the number of iterations isincreased, a full DoF in K user channel cannot be achievable since itcauses time delay that results in significant interference mis-alignmentfor fast fading. To compensate for these drawbacks, we suggest the starfeedback method illustrated in FIG. 4.

FIG. 4 shows an example of star feedback method (method 3).

Referring FIG. 4, wireless network comprises a CSI-base station, aplurality of transmitters, a plurality of receivers. The wirelessnetwork using the star feedback method comprises an agent, called theCSI base station (CSI-BS) which collects CSI from all receivers,computes all beamformers using IA condition in equation 7 or equation 13and sends them back to corresponding transmitters. This method 3 isfeasible since the computation of beamformers for IA is linked with eachother and same interference matrices are commonly required for thosebeamformers. In large K, the star method allows much smaller delaycompared with CSI exchange methods (method 1, method 2).

Star feedback method for IA in wireless network will be described.

1. Initialization: CSI-BS determines which interferers are aligned onthe same dimension at each receiver. Then, it informs each receiver ofthe required channel information for computing IA.

2. Computation of v^([1]), . . . , v^([K]): The CSI-BS collects CSI fromall receivers that comprises K×M² nonzero coefficients and computesbeamformers v^([1]), . . . , v^([K]) with the collected set of CSI.

3. Broadcasting v^([1]), . . . , v^([K]): CSI-BS forwards v^([k]) totransmitter k for k=1, . . . , K, which requires M nonzero coefficientto each of K transmitters.

For the star feedback method in MIMO channel, the network overhead isgiven as below equation.N _(SF)=(M ² +M)K  [equation 15]

Star feedback method requires only two time slots for computation of IAbeamformers in any number of user K. Therefore, it is robust againstchannel variations due to the fast fading while CSI exchange methods areaffected by 2K−1 slot delay for implementation. However, the networkoverhead of star feedback method is increased with KM² which is largerthan 2KM in modified CSI exchange method. Also, star feedback methodrequires CSI-BS that connects all pairs of transmitter-receiver shouldbe implemented as the additional costs.

The Network overhead of star feedback method can be quantified as below.

i) The overhead for initialization: Each receiver has K−1 interferersfrom other transmitters. Therefore, _(k−1)C₂ groups of alignedinterferers exist at each receiver, where _(k−1)C₂=(K−1)(K−2)/2. Toinform each receiver about the group of aligned interferers, totalKlog₂((K−1)(K−2)/2) bits are required for initial signaling.

ii) Feedback overhead for star feedback method: The CSI-BS collects CSIof product channels from all receivers that comprises KM² nonzerocoefficients. After computing the precoding vectors, CSI-BS transmits aprecoder of M nonzero coefficient to each of K transmitters. Therefore,(KM²+2M) nonzero coefficients are required for feedback in star feedbackmethod.

The above result shows that the network overhead for the star feedbackmethod is a linear function of K in contrast with the cubic function forconventional feedback method in MIMO channel. Thus the former methodleads to a much slower increase of network overhead with K.

III. Effect of CSI Feedback Quantization

In the preceding description, the CSI feedback methods are designed onthe assumption of perfect CSI exchange. However, CSI feedback fromreceiver to transmitter requires the channel quantization under thefinite-rate feedback constraints in practical implementation. Thisquantized CSI causes the degradation of system performance due to theresidual interference at each receiver. In this section, we characterizethe throughput loss as the performance degradation in limited feedback.Furthermore, we derive an upper-bound of sum residual interference asthe throughput loss and analyze it as a function of the number offeedback bits in given channel realization.

A. Throughput Loss in Limited Feedback

For the analytical simplicity, let define the throughput loss,ΔR _(sum)

as below equation.

$\begin{matrix}\begin{matrix}{{\Delta\; R_{sum}}:={E_{H}\left\lbrack {{\sum\limits_{k = 1}^{K}{\log_{2}\left( \frac{{Pd}_{kk}^{- \alpha}{{r^{\lbrack{k:\dagger}}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{\sigma^{2}} \right)}} - {\sum\limits_{k = 1}^{K}{\log_{2}\left( \frac{{Pd}_{kk}^{- \alpha}{{r^{{\lbrack k\rbrack}\dagger}H^{\lbrack{kk}\rbrack}v^{\lbrack k\rbrack}}}^{2}}{{\hat{I}}^{\lbrack k\rbrack} + \sigma^{2}} \right)}}} \right\rbrack}} \\{= {{E_{H}\left\lbrack {\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}} \right\rbrack}\overset{(a)}{\leq}{E_{H}\left\lbrack {K\;{\log_{2}\left( {\frac{1}{K}{\sum\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}}} \right)}} \right\rbrack}}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

In equation 16,{circumflex over (v)} ^([k])and{circumflex over (r)} ^([k])are the transmit beamformer and receive filter based on the quantizedCSI and ‘(a)’ follows from the characteristic of concave functionlog(x), denoting

${{{{{\hat{I}}^{\lbrack k\rbrack} = {\sum\limits_{{j = 1},{j \neq k}}^{K}\frac{P}{d_{kj}^{\alpha}}}}}{\hat{r}}^{{\lbrack k\rbrack}\dagger}H^{\lbrack{kj}\rbrack}{\hat{v}}^{\lbrack j\rbrack}}}^{2}$

Applying Jensen's inequality, the throughout loss is upper-bounded bybelow equation.

$\begin{matrix}{{\Delta\; R_{sum}} \leq {K\;{\log_{2}\left( {\frac{1}{K}{E_{H}\left\lbrack {\sum\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right\rbrack}} \right)}}} & \left\lbrack {{equation}\mspace{14mu} 17} \right\rbrack\end{matrix}$

In equation 17,ΔR _(sum)

is significantly affected by the sum residual interference

${\hat{I}}^{sum} = {\sum\limits_{k = 1}^{K}{{\hat{I}}^{\lbrack k\rbrack}.}}$Therefore, the minimization of sum rate loss is equivalent to theminimization of sum residual interference caused by limited feedback bitconstraints.

B. Residual Interference in the Proposed CSI Feedback Methods

Prior to deriving the bound of sum residual interference in proposed CSIfeedback methods, we quantify the quantization error with RVQ using thedistortion measure.

Let denote h:=vec(H)/∥H∥,ĥ:=vec({circumflex over (H)})/∥Ĥ∥ _(F)

and the phase rotation,e ^(jφ) :=ĥ ^(†) h/|ĥ ^(†) h|,whereH,HεC ^(N×M).

Then, h can be expressed by below equation.h−e ^(jφ) ĥ+Δh  [equation 18]

In equation 18,Δhrepresents the difference between h ande ^(jφ) ĥ.H is follows below equation.ΔHis defined such thatvec(ΔH)−Δh.H=e ^(jφ) Ĥ|ΔH

Lemma 3. The expected squared norm of the random matrixΔh

is bounded as below equation.

$\begin{matrix}{{E\left\lbrack {{\Delta\; H}}_{F}^{2} \right\rbrack} \leq {2\;{{\overset{\_}{\Gamma}({MN})} \cdot 2^{- \frac{B}{{MN} - 1}}}}} & \left\lbrack {{equation}\mspace{14mu} 20} \right\rbrack\end{matrix}$

In equation 20, B denotes quantization bit for H and

${\overset{\_}{\Gamma}({MN})} = {\frac{2\;{\Gamma\left( \frac{1}{{MN} - 1} \right)}}{{MN} - 1}.}$

Using properties of lemma 3, we rewrite equation 18 and equation 19 asbelow equation.h=e ^(jφ) ĥ+σ _(Δh) Δ{tilde over (h)}  [equation 21]H−e ^(jφ) Ĥ+σ _(Δh) Δ{tilde over (H)}{tilde over (,)}  [equation 21]

In equation 21,Δh−σ _(Δh) Δ{tilde over (h)},E[∥Δ{tilde over (h)}∥ ²]−1,E[∥Δ{tilde over(H)}∥ ² _(F)]−1

and

$\sigma_{\Delta\; h}^{2} \leq {{\overset{\_}{\Gamma}({MN})} \cdot {2^{- \frac{B}{{MN} - 1}}.}}$

Also, we derive the error bound of eigenvector of the quantized matrix,which is applied to analyze the sensitivity towards quantization erroron the initialization of v^([1]). Based on the modeling of quantizationerror of H in equation 22, we derive the eigenvector ofĤusing the perturbation theory in following lemma.

Lemma 4. For a large B, the m-th eigenvector{circumflex over (v)} _(m)ofĤ

is given as below equation.

$\begin{matrix}{{\hat{v}}_{m} = {{\mathbb{e}}^{{- j}\;\varphi}\left( {v_{m} - {\sigma_{\Delta\; h}{\sum\limits_{{k = 1},{k \neq m}}^{M}{\frac{v_{k}^{*}\Delta\;\overset{\sim}{H}v_{m}}{\lambda_{m} - \lambda_{k}}v_{k}}}}} \right)}} & \left\lbrack {{equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

In equation 22, v_(m) and λ_(m) are the m-th eigenvector and eigenvalueof H.

1) Modified CSI exchange method: As the modified CSI exchange methodrequires smaller sum overhead than that of CSI exchange method, wederive the upper bound of residual interference in modified CSI exchangemethod that consists of two types of feedback channel links: i) Feedbackof the channel matrix for the initial v^([1])and ii) Sequential exchangeof the quantized beamformer between transmitter and receiver.

For the initialization of v^([1]), the K-th receiver transmits theproduct channel matrix H eff^(K) to the (K−1)th receiver for thecomputation of v^([1]) which is designed as the eigenvector of H_(eff)^([K−1])H_(eff) ^([K]), where

$H_{eff}^{\lbrack{K - 1}\rbrack}:{- \frac{\left( H^{\lfloor{K - 11}\rfloor} \right)^{1}H^{\lfloor{K - 12}\rfloor}}{{{\left( H^{\lbrack{K - 11}\rbrack} \right)^{- 1}H^{\lbrack{K - 12}\rbrack}}}_{F}}}$and$H_{eff}^{\lbrack K\rbrack}:={\frac{\left( H^{\lbrack{K\; 2}\rbrack} \right)^{- 1}H^{\lbrack{K\; 1}\rbrack}}{{{\left( H^{\lceil{K\; 2}\rceil} \right)^{- 1}H^{\lceil{K\; 1}\rceil}}}_{F}}.}$

Assuming that RVQ is applied and the quantized matrixĤ ^([K]) _(eff)

is transmitted through B_(initial) bits feedback channel, the quantizedchannel matrix is expressed as below equation.Ĥ ^([K]) _(eff) =e ^(−jφinitial)(H ^([K]) _(eff)−σ_(initial) ΔĤ ^([K])_(eff))

In equation 23,

$\sigma_{initial}^{2} \leq {{\overset{\_}{\Gamma}\left( M^{2} \right)}2^{- \frac{B_{initial}}{M^{2} - 1}}}$

andE[∥ΔĤ ^([K]) _(eff)∥² _(F)]=1.Then, K−1-th receiver computes the quantized beamformer{circumflex over (v)} ^([1])ofH _(eff) ^([K−1]) Ĥ _(eff) ^([K])as below equation.{circumflex over (v)} ^([1]) =e ^(−jφinitial)(v ^([1]) −Δv^([1]))  [equation 24]

In equation 24, the quantization errorΔv[ ^(1])

is expressed by below equation.

$\begin{matrix}{{\Delta\; v^{\lbrack 1\rbrack}} = {\sigma_{initial}{\sum\limits_{{k = 1},{k \neq m}}^{M}{\frac{v_{k}^{*}H_{eff}^{\lfloor{K - 1}\rfloor}\Delta\;{\hat{H}}_{eff}^{\lfloor K\rfloor}v_{m}}{\lambda_{m} - \lambda_{k}}v_{k}}}}} & \left\lbrack {{equation}\mspace{14mu} 25} \right\rbrack\end{matrix}$

In equation 25, v_(m) and λ_(i) are the m-th eigenvector and eigenvalueof H_(eff) ^([K−1])H_(eff) ^([K]).

Since the magnitude of quantization error∥Δv ^([1])∥²

is represented as below equation, the quantization error of limitedfeedback is affected by exponential function of B_(initial) and thedistribution of eigenvalues |λ_(m)−λ_(k)|.

$\begin{matrix}{{{\Delta\; v^{\lbrack 1\rbrack}}}^{2} \propto {\sum\limits_{{k = 1},{k \neq m}}^{M}\frac{\sigma_{initial}^{2}}{{{\lambda_{m} - \lambda_{k}}}^{2}}}} & \left\lbrack {{equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

Secondly, the effect of quantization error due to the exchange ofbeamformers between transmitters and receivers is described. On theassumption of perfect CSI of H eff^([K]) at receiver K−1, it isconsidered the limited feedback links from receiver to transmitter,where B_(k) bits are allocated to quantize{circumflex over (v)} ^([k]).

From modified CSI exchange method (method 2), receiver K−1 computesv^([1]) and quantize it to{circumflex over (v)} ^([1])

with RVQ and feeds back to transmitter 1 through B₁ feedback channel.The quantized beamformer{circumflex over (v)} ^([1])

is represented as below equation.{circumflex over (v)} ^([1]) −e ^(−jφK−1)(v ^([1])−σ₁ Δv^([1]))  [equation 27]

In equation 27,σ₁ ²= Γ(M)2−_(M1) ^(D) ¹andE[∥Δv ^([1])∥²]=1.

Following the procedure of method 2, transmitter 1 forwards{circumflex over (v)} ^([1])

to receiver K. The receiver K designs{dot over (v)} ^([2])

on the subspace ofH _(eff) ^([K]) {circumflex over (v)} ^([1])

as below equation.

$\begin{matrix}\begin{matrix}{{\overset{.}{v}}^{\lbrack 2\rbrack} = {H_{eff}^{\lbrack K\rbrack}{\hat{v}}^{\lbrack 1\rbrack}}} \\{= {{\mathbb{e}}^{{- j}\;\varphi_{1}}\left( {{H_{eff}^{\lbrack K\rbrack}v^{\lbrack 1\rbrack}} - {\sigma_{1}H_{eff}^{\lbrack K\rbrack}\Delta\; v^{\lbrack 1\rbrack}}} \right)}} \\{= {{\mathbb{e}}^{{- j}\;\varphi_{1}}\left( {v^{\lbrack 2\rbrack} - {\sigma_{1}H_{eff}^{\lbrack K\rbrack}\Delta\; v^{\lbrack 1\rbrack}}} \right)}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 28} \right\rbrack\end{matrix}$

Then, receiver K quantizes{dot over (v)} ^([2])

is quantized to{circumflex over (v)} ^([2])

with B₂ bit quantization level and informed to transmitter 2. Thequantized precoder{circumflex over (v)} ^([2])

is represented as below equation.{circumflex over (v)} ^([2]) =e ^(−jφ2)({dot over (v)} ^([2])−σ₂ Δv^([2]))  [equation 29]

In equation 29,

$\sigma_{2}^{2} = {{\overset{\_}{\Gamma}(M)}2^{- \frac{B_{2}}{M - 1}}}$

andE[∥Δv ^([2])∥²]=1.

Likewise, the other beamformers{dot over (v)} ^([3]) ,{dot over (v)} ^([4]), . . . , and {dot over (v)}^([K])are designed at receiver 1, 2, . . . , and K−2 and their quantizedbeamformers{circumflex over (v)} ^([2]) ,{circumflex over (v)} ^([4]), . . . , and{circumflex over (v)} ^([K])are fed back to their corresponding transmitters, where{circumflex over (v)} ^([k])is modeled as below equation.v ^([k]) =e ^(−jφk)({dot over (v)} ^([k])−σ_(k) Δv ^([k]))  [equation30]

In equation 30,

$\sigma_{k}^{2} = 2^{- \frac{B_{k}}{M - 1}}$

andE[∥Δv ^([k])∥²]=1,

for k=3, 4, . . . , K.

Using equation 28, 29 and 30, the sum residual interference affected byquantization error can be analyzed. To analyze the residual interferenceat each receiver, it is assumed that each receiver designs azero-forcing receiver with a full knowledge of{v ^([k]):1≦k≦K},

which cancels M−1 dimensional interferers. Then, the upper-bound ofexpected sum residual interference is expressed as a function of thefeedback bits, the eigenvalue of fading channel and distance between thepairs of transmitter-receiver is suggested as below.

Proposition 3

In modified CSI feedback method, the upper-bound of expected residualinterference at each receiver is represented as below equation in givenfeedback bits {B_(k)}_(k=1) ^(K) and channel realization{H^([jk])}_(j,k=1) ^(K).

                                [equation  31] $\left\{ \begin{matrix}{{{E\left\lbrack {\hat{I}}^{\lfloor k\rfloor} \right\rbrack} \leq {{Pd}_{{kk} + 2}^{- \alpha}\sigma_{k + 2}^{2}\lambda_{\max}^{\lbrack{{kk} + 2}\rbrack}}},\mspace{11mu}{{{for}\mspace{14mu} k} = 1},\ldots\mspace{14mu},{K - 2}} \\{{E\left\lbrack {\hat{I}}^{\lbrack{K - 1}\rbrack} \right\rbrack} \leq {{Pd}_{K - 11}^{- \alpha}\left( {\sigma_{2}^{2}\lambda_{\max}^{\lbrack{K - 12}\rbrack}{{\sigma_{1}^{2}\lambda_{\max}^{\lbrack{K - 12}\rbrack}}}\sigma_{1}^{2}\lambda_{\max}^{\lbrack{K - 11}\rbrack}} \right)}} \\{{E\left\lbrack {\hat{I}}^{\lbrack K\rbrack} \right\rbrack} \leq {{Pd}_{K\; 2}^{- \alpha}\sigma_{2}^{2}\lambda_{\max}^{\lbrack{K\; 2}\rbrack}}}\end{matrix} \right.$

2) Star feedback method: In star feedback method, it is assumed that allreceivers are connected to CSI-BS with high capacity backhaul links,such as cooperative multicell networks. Then, CSI-BS is allowed toacquire full knowledge of CSI estimated at each receiver. Based onequation 13, CSI-BS computes v^([1]), v^([2]), . . . , v^([K]) andforwards them to the corresponding transmitters. In this section, weconsider the B_(k) feedback bits constraint from CSI-BS to transmitter kand v^([k]) is quantized as below equation.{circumflex over (v)} ^([k]) =e ^(−jφk)(v ^([k])−σ_(k) Δv^([k]))  [equation 32]

In equation 32,

$\sigma_{k}^{2} = {{\overset{\_}{\Gamma}(M)}2^{- \frac{B_{k}}{M - 1}}}$

andE[∥Δv ^([k])∥²]=1.

Given the quantized beamformer, each transmitter sends a data streamthat causes the residual interference at the receiver. Followingproposition provides the upper-bound of expected residual interferenceat each receiver that consists of the feedback bits, the eigenvalue offading channel and distance between the pairs of transmitter-receiver.

Proposition 4

In star feedback method, the upper-bound of expected residualinterference at each receiver is represented in given feedback bits{B_(k)}_(k=1) ^(K) and channel realization {H^([jk])}_(j,k=1) ^(K) belowequation.

                                     [equation  33]$\left\{ \begin{matrix}{{{{E\left\lbrack {\hat{I}}^{\lbrack k\rbrack} \right\rbrack} \leq {{{Pd}_{{kk} + 2}^{- \alpha}\sigma_{k + 1}^{2}\lambda_{\max}^{\lbrack{{kk} + 1}\rbrack}} + {{Pd}_{{kk} + 2}^{- \alpha}\sigma_{k + 2}^{2}\lambda_{\max}^{\lbrack{{kk} + 2}\rbrack}\mspace{14mu}{for}\mspace{14mu} k}}} = 1},\ldots\mspace{14mu},{K - 2}} \\{{E\left\lbrack {\hat{I}}^{\lbrack{K - 1}\rbrack} \right\rbrack} \leq {{{Pd}_{K - 11}^{- \alpha}\sigma_{1}^{2}\lambda_{\max}^{\lbrack{K - 11}\rbrack}} + {{Pd}_{K - 11}^{- \alpha}\sigma_{2}^{2}\lambda_{\max}^{\lbrack{K - 12}\rbrack}}}} \\{{E\left\lbrack {\hat{I}}^{\lbrack K\rbrack} \right\rbrack} \leq {{{Pd}_{K\; 2}^{- \alpha}\sigma_{1}^{2}\lambda_{\max}^{\lbrack{K\; 1}\rbrack}} + {{Pd}_{K\; 2}^{- \alpha}\sigma_{2}^{2}\lambda_{\max}^{\lbrack{K\; 2}\rbrack}}}}\end{matrix} \right.$

IV. Dynamic Feedback Bit Allocation Methods for CSI Feedback Methods

The system performance of IA is significantly affected by the number offeedback bits that induces residual interference in a finite ratefeedback channel. The dynamic feedback bit allocation strategies thatminimize the throughput loss due to the quantization error in given sumfeedback bits constraint is explained. Furthermore, It is provided therequired number of feedback bits achieving a full network DoF in eachCSI feedback methods.

A. Dynamic Bit Allocation for Minimizing Throughput Loss

Consider total B_(tot) bits are given for the feedback framework in thenetwork model. Since the throughput loss is bounded by the expected sumresidual interference, we can formulate the dynamic bit allocationproblem for minimizing throughput loss as follows equation.

$\begin{matrix}{{\min\mspace{14mu}{\sum\limits_{k = 1}^{K}{E\left\lbrack {\hat{I}}^{\lbrack k\rbrack} \right\rbrack}}}{{s.t.\mspace{14mu}{\sum\limits_{k = 1}^{K}B_{k}}} \leq B_{tot}}} & \left\lbrack {{equation}\mspace{14mu} 34} \right\rbrack\end{matrix}$

Using proposition 3 and proposition 4, equation 34 can be transformed toconvex optimization problem with variables {B_(k)}_(k=1) ^(K) expressedby below equation.

$\begin{matrix}{{\min\mspace{14mu}{\sum\limits_{k = 1}^{K}{a_{k}2^{-_{M\mspace{11mu} 1}^{B_{k}}}}}}{{s.t.\mspace{14mu}{\sum\limits_{k = 1}^{K}B_{k}}} \leq B_{tot}}} & \left\lbrack {{equation}\mspace{14mu} 35} \right\rbrack\end{matrix}$

Here, we define a_(k) in modified CSI exchange method (method 2) asbelow equation.

                                [equation  36] $\begin{matrix}{a_{k} = \left\{ \begin{matrix}{a_{1} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K - 11}^{- \alpha}\lambda_{\max}^{\lbrack{K - 12}\rbrack}}❘{d_{K - 11}^{- \alpha}\lambda_{\max}^{\lbrack{K - 11}\rbrack}}} \right)}}} \\{a_{2} - {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K\; 2}^{- \alpha}\lambda_{\max}^{\lbrack{K\; 2}\rbrack}} + {d_{K - 11}^{- \alpha}\lambda_{\max}^{\lbrack{K - 12}\rbrack}}} \right)}}} \\{{a_{k} = {{{{\overset{\_}{\Gamma}(M)} \cdot {Pd}_{k - {2\; k}}^{- \alpha}}\lambda_{\max}^{\lbrack{k - {2\; k}}\rbrack}\mspace{25mu}{\forall k}} = 3}},\ldots\mspace{11mu},K}\end{matrix} \right.} & \;\end{matrix}$

And, a_(k) in star feedback method as below equation.

                                     [equation  37]$a_{k} = \left\{ \begin{matrix}{a_{1} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K\; 2}^{\alpha}\lambda_{\max}^{\lbrack{K\; 1}\rbrack}} + {d_{K - 11}^{- \alpha}\lambda_{\max}^{\lbrack{K - 11}\rbrack}}} \right)}}} \\{a_{2} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{13}^{- \alpha}\lambda_{\max}^{\lbrack 12\rbrack}} + {d_{K\; 2}^{- \alpha}\lambda_{\max}^{\lbrack{K\; 2}\rbrack}}} \right)}}} \\{{a_{k} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{k - {1\; k} + 1}^{- \alpha}\lambda_{\max}^{\lbrack{k - {1\; k}}\rbrack}} + {d_{k - {2\; k}}^{{- \alpha}}\lambda^{\lbrack{k - {2\; k}}\rbrack}}} \right)}}},{k = 3},\ldots\mspace{14mu},{K - 1}} \\{a_{K} = {{\overset{\_}{\Gamma}(M)} \cdot {P\left( {{d_{K - 11}^{- \alpha}\lambda_{\max}^{\lbrack{K - {1\; K}}\rbrack}} + {d_{K - {2\; K}}^{- \alpha}\lambda_{\max}^{\lbrack{K - {2\; K}}\rbrack}}} \right)}}}\end{matrix} \right.$

Consider the objective function in equation 35. By forming thelagrangian and taking derivative with respect to B_(k), it can beexpressed by below equation.

$\begin{matrix}{{L = {{\sum\limits_{k \in U}{a_{k}2^{- \frac{B_{k}}{M - 1}}}} + {v\left( {{\sum\limits_{k \in U}B_{k}} - B_{tot}} \right)}}}{\frac{\partial L}{\partial B_{k}} = {{{- 2^{- \frac{B_{k}}{M - 1}}}\ln\; 2\frac{a_{k}}{M - 1}} + v - 0}}} & \left\lbrack {{equation}\mspace{14mu} 38} \right\rbrack\end{matrix}$

In equation 38, v is the Lagrange multiplier and U is the set offeedback link U={1, . . . K}. Therefore, we obtain B_(k) as belowequation.

$\begin{matrix}{B_{k} = {{\left( {M - 1} \right) \cdot \log}\; 2\left( \frac{\mu\; a_{k}}{M - 1} \right)}} & \left\lbrack {{equation}\mspace{14mu} 39} \right\rbrack\end{matrix}$

and B_(k) satisfies the following constraint equation where

$\mu = {\frac{\ln\; 2}{v}.}$

$\begin{matrix}{{\sum\limits_{k \in U}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{\mu\; a_{k}}{M - 1} \right)}}} = B_{tot}} & \left\lbrack {{equation}\mspace{14mu} 40} \right\rbrack\end{matrix}$

Combining equation 39 and 40 with B_(k)≧0, the number of optimalfeedback bit is obtained as below equation.

$\begin{matrix}{{B_{k}^{*} = {\frac{1}{U}\left( {\gamma - {\left( {M - 1} \right) \cdot {U} \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}} \right)^{+}}}{\mu = \left( {2^{B_{tot}}{\prod\limits_{k \in U}\;\left( \frac{M - 1}{a_{k}} \right)^{M - 1}}} \right)^{\frac{1}{{({M - 1})} \cdot {U}}}}} & \left\lbrack {{equation}\mspace{14mu} 41} \right\rbrack\end{matrix}$

In equation 41,

$\gamma = {B_{tot} + {\sum\limits_{k \in U}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}}}$|U| is the cardinality of U and

$(a)^{+} = \left\{ \begin{matrix}a & {{{if}\mspace{14mu} a} \geq 0} \\0 & {{{if}\mspace{14mu} a} < 0.}\end{matrix} \right.$

The solution of equation 41 is found iteratively through thewaterfilling algorithm, which is described as bellow.

Waterfilling algorithm.

--------------------------------------------------------i=0; U={1, . . . ,K};

while i=0 do

Determine the water-level

$\gamma = {B_{tot}❘{\sum\limits_{k \in U}{\left( {M - 1} \right) \cdot {\log_{2}\left( \frac{M - 1}{a_{k}} \right)}}}}$

Choose the user set

$\overset{\_}{k} = {\arg\;\max{\left\{ {\frac{M - 1}{a_{k}}:{k \in U}} \right\}.{if}}}$${\gamma - {\left( {M - 1} \right) \cdot {U} \cdot {\log_{2}\left( \frac{M - 1}{a_{\overset{\_}{k}}} \right)}}} \geq 0$

then optimal bit allocationB* _(k)in U is determined by equation 43.i=i+1;

else Let defineU−{U except for k } and

$B_{\overset{\_}{k}}^{*} - 0$

------------------------------------------------------------

From the waterfilling algorithm, we obtain {B_(k)*:kεU}. However, B_(k)*should become integer so that it is determined the optimal bitallocationB* _(k)

as below equation.B* _(k) =└B* _(k)┘  [equation 42]

In equation 42,└x┘

is the largest integer not greater than x.

B. Scaling Law of Total Feedback Bits

In perfect CSI assumption, each pair of transmitter-receiver linkobtains the interference-free link for its desired data stream. However,misaligned beamformers due to the quantization error destroy the linearscaling gain of sum capacity at high SNR regime. In this section, weanalyze the total number of feedback bits that achieve a full networkDoF K in the proposed CSI feedback methods. The required sum feedbackbits are formulated as the function of the channel gains, the number ofantennas and SNR that maintain the constant sum rate loss over the wholeSNR regimes.

As P goes to infinity, the network DoF in K user interference channelachieves K with constant value of

$\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}$

according to below equation.

$\begin{matrix}\begin{matrix}{{DoF} = {\lim\limits_{P->\infty}\frac{R_{sum}^{limited}}{\log_{2}P}}} \\{= {\lim\limits_{P->\infty}\frac{\begin{matrix}{{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\frac{P}{d_{kk}^{\alpha}}{{{\hat{r}}^{\lbrack k\rbrack}H^{\lbrack{kk}\rbrack}{\hat{v}}^{\lbrack k\rbrack}}}^{2}} \right)}} -} \\{\sum\limits_{i = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}}\end{matrix}}{\log_{2}P}}} \\{= {{\lim\limits_{P->\infty}\frac{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\begin{matrix}P \\d_{kk}^{\alpha}\end{matrix}{{{\hat{r}}^{\lbrack k\rbrack}H^{\lbrack{kk}\rbrack}{\hat{v}}^{\lbrack k\rbrack}}}^{2}} \right)}}{\log_{2}P}} -}} \\{\lim\limits_{P->\infty}\frac{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}}{\log_{2}P}} \\{= {K - {\lim\limits_{P->\infty}\;\frac{\log_{2}\left( {\prod\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right)}{\log_{2}P}} - K}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 43} \right\rbrack\end{matrix}$

Therefore, it can be formulated the sum residual interference as anexponential function of B_(k) as below equation.

$\begin{matrix}\begin{matrix}{{\sum\limits_{k = 1}^{K}{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)}} = {\log_{2}\left( {\prod\limits_{k = 1}^{K}{\hat{I}}^{\lbrack k\rbrack}} \right)}} \\{= {\log_{2}\left( {\prod\limits_{k = 1}^{K}{a_{k}2^{- \frac{B_{k}}{M - 1}}}} \right)}} \\{= c}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 44} \right\rbrack\end{matrix}$

Equation 44 yields an equation 45.

$\begin{matrix}\begin{matrix}{B_{tot}^{*} = {\sum\limits_{k = 1}^{K}B_{k}}} \\{= {\left( {M - 1} \right) \cdot \left( {{\overset{K}{\sum\limits_{k}}{\log_{2}a_{k}}} - {\log_{2}c}} \right)}} \\{= {{{K \cdot \left( {M - 1} \right) \cdot \log_{2}}P} + {\left( {M - 1} \right) \cdot \left( {{\sum\limits_{k - 1}^{K}{\log_{2}{\hat{a}}_{k}}} - c} \right)}}}\end{matrix} & \left\lbrack {{equation}\mspace{20mu} 45} \right\rbrack\end{matrix}$

In equation 45, c is constant and c>0 anda _(k) =P·â _(k) ,∀k.

Since B_(tot)* is the integer number, we obtain the required feedbackbitsB* _(tot)

for achieving full DoF as below equation.B* _(tot) =nint(B* _(tot))

In equation 46, nint(x) is the nearest integer function of x.

C. Implementation of Feedback Bits Controller

As we derived in equation 41 and 45, the computation of{B* _(k)}^(K) _(k−1)andB _(tot)requires the set of channel gain {a_(k)}_(k=1) ^(K) that consist of thepath-loss and short-term fading gain from all receivers. Therefore, theoptimal bit allocation strategy requires the centralized bit controllerthat gathers full knowledge of {a_(k)}_(k=1) ^(K) and computes theoptimal set of feedback bits. The centralized bit controller can befeasible in star feedback method since it has a CSI-BS connected withall receivers through the backhaul links.

However, the receiver in CSI exchange method only feeds back CSI tocorresponding transmitter, implemented by distributedfeedforward/feedback channel links. Therefore, the additional bitcontroller is required to allocate optimal feedback bits in CSI exchangemethod. Moreover, {a_(k)}_(k=1) ^(K) includes the gain of short-termfading so that CSI of all receivers should be collected to centralcontroller over every transmission period, which requires a largeoverhead of CSI exchange and causes delay that results in significantinterference misalignment in fast channel variation environments.

1) Path-loss based bit allocation method: To reduce the burden offrequent exchange of channel gains for bit allocation, we average{a_(k)}_(k=1) ^(K) over the short-term fading {H_([ij])}_(i,j=1) ^(K).Consider

$\begin{matrix}{{E_{H}\left\lbrack a_{k} \right\rbrack} = \left\{ \begin{matrix}{{E_{H}\left\lbrack a_{1} \right\rbrack} = {{{\overset{\_}{\Gamma}(M)} \cdot 2}{P \cdot M^{2} \cdot d_{K - 11}^{- \alpha}}}} \\{{E_{H}\left\lbrack a_{2} \right\rbrack} - {{\overset{\_}{\Gamma}(M)} \cdot {PM}^{2} \cdot \left( {d_{K\; 2}^{- \alpha} + d_{K - 11}^{- \alpha}} \right)}} \\{{{E_{H}\left\lbrack a_{k} \right\rbrack} = {{{\overset{\_}{\Gamma}(M)} \cdot {PM}^{2} \cdot d_{k - {2k}}^{- \alpha}}k}},{{\forall k} = 3},\ldots\mspace{14mu},K}\end{matrix} \right.} & \left\lbrack {{equation}\mspace{14mu} 47} \right\rbrack\end{matrix}$

and apply it to equation 36. E_(H)[a_(k)] in modified CSI exchangemethod is represented as below equation where the expected value of ak'sconsists of transmit power, the number of antennas and path-loss.

E[λ_(ma x)^([ij])] ≤ M²

Applying equation 47 to 41 and 45, the dynamic bit allocation shceme canbe implemented by the path-loss exchange between receivers. Since thepath-loss shows a long-term variability compared with fast fadingchannel, the additional overhead for bit allocation is required over themuch longer period than the bit allocation scheme in equation 41 and 45.

2) Distributed bit allocation scheme in CSI exchange method: We developthe distributed bit allocation scheme achieving DoF K in modified CSIexchange method, without centralized bit controller. For the distributedbit allocation, we replace the condition (equation 44) that satisfiesDoF K as below equation.

$\begin{matrix}\begin{matrix}{{\log_{2}\left( {\hat{I}}^{\lbrack k\rbrack} \right)} = {\log_{2}\left( {a_{k}2^{- \frac{B_{k}}{M - 1}}} \right)}} \\{{= \frac{c}{K}},{{\forall k} = 1},\ldots\mspace{14mu},K}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 48} \right\rbrack\end{matrix}$

Therefore, the required feedback bit Bk for k-th beamformer is derivedas below equation where a_(k) is represented in equation 36.

$\begin{matrix}{B_{k} - {{\left( {M - 1} \right) \cdot \log_{2}}a_{k}} - {{\left( {M - 1} \right) \cdot \log_{2}}\frac{c}{K}}} & \left\lbrack {{equation}\mspace{14mu} 49} \right\rbrack\end{matrix}$

Combining equation 49 with 36, each receiver can compute the requiredfeedback bits with its own channel knowledge except for receiver K.Receiver K requires additional knowledge of

d_(K − 11)^(−α)λ_(ma x)^([K − 12])

from receiver K−1 to calculate a₂. From equation 49, it is provided thedesign of modified CSI exchange method with bit allocation of achievingIA in distributed manners as below.

1. step 1; set-up: Receiver K forwards the matrix (H^([K2]))⁻¹H^([K1])to receiver K−1.

2. step 2; computation of v^([1]): Receiver K−1 computes v^([1]) as anyeigenvector of H^([K−11]))⁻¹H^([K−12])(H^([K2]))⁻¹H^([K1]) and quantizesit to{circumflex over (v)} ^(|1|)

with B₁ computed by equation 49.{circumflex over (v)} ^([1])

is forwarded to transmitter 1. Also, receiver K−1 forwards

d_(K − 11)^(−α)λ_(m ax)^([K − 12])

to receiver K.

3. step 3; computation of v[2]: Transmitter 1 feeds forward{circumflex over (v)} ^([1])

to receiver K. Then, receiver K calculates{dot over (v)} ^([2])

and quantizes it to{circumflex over (v)} ^([2])

with B₂ computed by equation 49.{circumflex over (v)} _([2])

is fed back to transmitter 2.

4. step 4; computation of v^([3]), . . . , v^([K])

For i=3: K, Transmitter i−1 forwards{circumflex over (v)} ^([i−1])to receiver i−2. Then, receiver i−2 calculates{dot over (v)} ^([i])and quantizes it to{circumflex over (v)} ^([i])with B_(i) computed by equation 49.{circumflex over (v)} ^([i])is fed back to transmitter i−1.

V. Successive CSI Exchange Method Based on Precoded RS

CSI exchange method in previous description can reduce the amount ofnetwork overhead for achieving IA. However, it basically assumes thefeedforward/feedback channel links to exchange pre-determinedbeamforming vectors.

Considering precoded RS in LTE (long term evolution), the CSI exchangemethod can be slightly modified for the exchange of precoding vector(i.e. beamformer or beamforming vector). However, it still effectivelyscales down CSI overhead compared with conventional feedback method.

Consider K=M+1 user interference channel, where each transmitter andreceiver are equipped with M antennas and DoF(d) for each user is d=1.Based on IA condition and precoding vector in equation 6 and equation 7,we firstly explain the procedure of CSI exchange method infeedforward/feedback framework for K=4. After that, we describe theprocedure of CSI exchange method based on precoded RS, in detail.

1. Successive CSI Exchange in Method 1(e.g. K=4)

In K=4, IA condition is shown as below equation.

$\begin{matrix}{\begin{matrix}{{{span}\left( {H^{\lbrack 13\rbrack}V^{\lbrack 3\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 14\rbrack}V^{\lbrack 4\rbrack}} \right)}\mspace{14mu}{At}\mspace{14mu}{receiver}\mspace{14mu} 1}} \\{{{span}\left( {H^{\lbrack 21\rbrack}V^{\lbrack 1\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 24\rbrack}V^{\lbrack 4\rbrack}} \right)}\mspace{14mu}{At}\mspace{14mu}{receiver}\mspace{14mu} 2}} \\{{{span}\left( {H^{\lbrack 31\rbrack}V^{\lbrack 1\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 32\rbrack}V^{\lbrack 2\rbrack}} \right)}\mspace{14mu}{At}\mspace{14mu}{receiver}\mspace{14mu} 3}} \\{{{span}\left( {H^{\lbrack 42\rbrack}V^{\lbrack 2\rbrack}} \right)} = {{{span}\left( {H^{\lbrack 43\rbrack}V^{\lbrack 3\rbrack}} \right)}\mspace{14mu}{At}\mspace{14mu}{receiver}\mspace{14mu} 4}}\end{matrix}->\begin{matrix}{{{span}\left( V^{\lbrack 1\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 31\rbrack} \right)^{- 1}H^{\lbrack 32\rbrack}V^{\lbrack 2\rbrack}} \right)}} \\{{{span}\left( V^{\lbrack 2\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 42\rbrack} \right)^{- 1}H^{\lbrack 43\rbrack}V^{\lbrack 3\rbrack}} \right)}} \\{{{span}\left( V^{\lbrack 3\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 13\rbrack} \right)^{- 1}H^{\lbrack 14\rbrack}V^{\lbrack 4\rbrack}} \right)}} \\{{{span}\left( V^{\lbrack 4\rbrack} \right)} = {{span}\left( {\left( H^{\lbrack 24\rbrack} \right)^{- 1}H^{\lbrack 21\rbrack}V^{\lbrack 1\rbrack}} \right)}}\end{matrix}} & \left\lbrack {{equation}\mspace{14mu} 50} \right\rbrack\end{matrix}$

Then, precoding vectors at transmitter 1, 2, 3 and 4 can be determinedas below equation.V ^([1])eigen vector of (H ^([31]))⁻¹ H ^([32])(H ^([42]))⁻¹ H ^([43])(H^([13]))⁻¹ H ^([14])(H ^([24]))⁻¹ H ^([21])V ^([2])=(H ^([32]))⁻¹ H ^([31]) V ^([1])V ^([3])=(H ^([43]))⁻¹ H ^([42]) v ^([2])V ^([4])=(H ^([14]))⁻¹ H ^([13]) V ^([3])  [equation 51]

In equation 50, the interference from transmitter 3 and 4 are on thesame subspace at receiver 1. Likewise, the interference from transmitter1 and 4, the interference from transmitter 1 and 2, and the interferencefrom transmitter 2 and 3 are on the same subspace at receiver 2,3, and4.

To inform the receiver that which interferers are aligned on the samedimension, CSI exchange method firstly requires initial signaling. Thisinitial signaling is transmitted with a longer period than that ofprecoder update.

After the initial signaling indicated which interferers are aligned ateach receiver, the receiver 1, 2, 3, 4 feed back the selectedinterference channel matrices multiplied by the inverse of anotherinterference channel matrices, {(H^([31]))⁻¹H^([32]),(H^([42]))⁻¹H^([43]), (H^([13]))⁻¹H^([14]), (H^([24]))⁻¹H^([21])} totransmitter 1. Based on these product channel matrices, transmitter 1determines V^([1]) and forwards it to receiver 3. Then, receiver 3determines V^([2]) using the channel information of(H^([32]))⁻¹H^([31]), and feeds it back to transmitter 2. With thisiterative way, V^([3]) and V^([4]) are sequentially determined. Thisprocess is already described in previous part. However, it is slightlydifferent in the point that V^([1]) is determined by transmitter 1 not areceiver 3.

FIG. 5 shows the procedure of CSI exchange method in case that K is 4.

Referring FIG. 5 (a), the procedure of CSI exchange method comprisesbelow steps.

step 1. Initial Signaling: Each transmitter (e.g. BS) reports whichinterferers are aligned on the same dimension to its receiver. (In K=4,two of four interferers are selected as the aligned interference.)

step 2. Determine initial vector V^([1]): After initialization, receiver1, 2, 3, 4 feed back the product channel matrices to transmitter 1, thentransmitter 1 can determine V^([1]).

step 3. Computation of V^([2]) V^([2]) and V^([4]): Transmitter 1 feedsforward V^([1]) to receiver 3 and receiver 3 calculates V^([2]). Afterthat, receiver 3 feeds back V^([2]) to transmitter 2. With the exchangeof pre-determined precoding vector, the remained precoding vectors aresequentially computed.

Referring FIG. 5 (b), the step 3 in the CSI exchange method is shown,where FF is feedfoward and FB is feedback links.

In a system using precoded RS such as a LTE system, each transmitter(e.g. BS) transmits the reference symbol with precoding vector.Therefore, the feedfoward signaling in successive CSI exchange can bemodified compared with above CSI exchange method.

FIG. 6 shows the procedure of CSI exchange method based on precoded RSin case that K is 4.

Assume that i-th receiver perfectly estimates the channel matrices{H^([1i]), H^([2i]), . . . , H^([Ki])}. Then precoding vectors V^([1]),V^([2]), V^([3]), and V^([4]) for IA are designed by equation 51 in K=4.To compute the precoding vectors, the procedure of CSI exchange methodbased on precoded RS is suggested as below.

Referring FIG. 6 (a), the procedure of CSI exchange method based onprecoded RS comprises below steps.

step 1. Initial signaling: Each BS (transmitter) reports whichinterferers are aligned on same dimension to its receiver. (The index ofaligned interferers is forwarded to the receiver.)

step 2. Determine initial vector V^([1]): After initialization, receiver1, 2, 3, 4 feed back the product channel matrices to transmitter 1, thentransmitter 1 can determine V^([1]). Here, the product channel matricescomprises (H^([31]))⁻¹H^([32]), (H^([42]))⁻¹H^([43]),(H^([13]))⁻¹H^([14]) and (H^([24]))⁻¹H^([21]).

step 3. computation V^([2]), V^([3]) and V^([4]).

referring FIG. 6( b), the process of the step 3 includes belowprocedures.

Transmitter 1: After determining its precoder V^([1]), transmitter 1sends the precoded RS to receiver 3.

Receiver 3: Receiver 3 estimates effective channel H_(eff)^([31])=H^([31])V^([1]). Based on equation 51, receiver 3 computesV^([2]) multiplying H_(eff) ^([31]) with (H^([32]))⁻¹. Then, receiver 3sends V^([2]) to transmitter 2 through feedback channel.

Transmitter 2: Transmitter 2 sends the precoded RS based on V^([2]) toreceiver 4.

Receiver 4: Receiver 4 estimates effective channel H_(eff)^([42])=H^([42])V^([2]) and it computes V^([3]) multiplying H_(eff)^([42]) with (H^([43]))⁻¹. Then, V^([3]) is reported to transmitter 3through feedback channel.

Transmitter 3: Transmitter sends the precoded RS based on V^([3]) toreceiver 1.

Receiver 1: Receiver 1 estimates effective channel H_(eff)^([13])=H^([13])V^([3]) and receiver 1 computes V^([4]) multiplyingH_(eff) ^([13]) with (H^([14]))⁻¹. Then, V^([4]) is reported totransmitter 4 through feedback channel.

VI. Efficient Interference Alignment Method for 3 Cell MIMO InterferenceChannel with Limited Feedback

There are known-closed form solutions for K user MIMO interferencechannel where K=3 with d=M/2 and K=M+1 with d=1. Here, d denotes adegree of freedom. In both case, each transmitter needs information of2K product channel matrices for the initialization in equation 6, 7since all precoders are coupled.

If quantized channel feedback is assumed, error of product-quantizedmatrices for IA solution is much larger than error of single channelmatrix quantization.

As an example, consider K=3 and M=2 with d=1. The solution of V^([1]) isany eigenvector of(H^([31]))⁻¹H^([32])(H^([12]))⁻¹H^([13])(H^([23]))⁻¹H^([21]). If eachreceiver feeds back a product channel matrix for aligned interferences(e.g at receiver 3, feeding back H_(eff) ^([3])=(H^([32]))⁻¹H^([31])H^([31]) to transmitter 1), the precoder at transmitter 1 is estimatedas below equation.

$\begin{matrix}\begin{matrix}{{\hat{V}}^{\lbrack 1\rbrack} = {{eig}\left( {\left( {{\left( H^{\lbrack 31\rbrack} \right)^{- 1}H^{\lbrack 32\rbrack}} + E^{\lbrack 3\rbrack}} \right)\left( {{\left( H^{\lbrack 12\rbrack} \right)^{- 1}H^{\lbrack 13\rbrack}} + E^{\lbrack 1\rbrack}} \right)\left( {{\left( H^{\lbrack 23\rbrack} \right)^{- 1}H^{\lbrack 21\rbrack}} + E^{\lbrack 2\rbrack}} \right)} \right)}} \\{= {{eig}\left( {{\left( H^{\lbrack 31\rbrack} \right)^{- 1}{H^{\lbrack 32\rbrack}\left( H^{\lbrack 12\rbrack} \right)}^{- 1}{H^{\lbrack 13\rbrack}\left( H^{\lbrack 23\rbrack} \right)}^{- 1}H^{\lbrack 21\rbrack}} + E_{total}} \right)}} \\{= {{eig}\left( {V^{\lbrack 1\rbrack} + E_{total}} \right)}}\end{matrix} & \left\lbrack {{equation}\mspace{14mu} 52} \right\rbrack\end{matrix}$

In equation 52, E^([n]) is channel quantization error at node n andE_(total) is all but V^([1]). On account of matrix multiplication,Frobenius norm of E_(total) is larger than it of E^([n]) since E_(total)is made up of summation and multiplication of E^([n]). We will refer tothis as the quantized error amplification (QEA) in interferencealignment system. In other words, to satisfy a given feedback CSIdistortion level in interference alignment system, much more feedbackbits are required than that of point to point network. Therefore, inlimited feedback system, IA solution for achieving the optimal DoF hasmuch degradation compared with perfect feedback. This demands thedevelopment of an efficient alignment method to avoid QEA.

Basically, the QEA is due to the chaining condition of IA solution. Mainidea of this embodiment of the present invention is use of fallowspatial DoFs to cut the chaining condition. If a node does not use somespatial DoF, IA solution is detached to several independent equations.

For example, let consider K=3 user and M×M MIMO interference channel. Itis assumed that transmitter 1 is the sacrifice node (any node can be thesacrifice node) which means the DoF for the transmitter 1(d₁) is M/2−α,where a is the fallow DoF, and other 2 nodes use d₂=d₃=M/2 DoF. Based onIA condition in equation 50, first M/2−α beams at transmitter 2, 3 haveto be aligned with beams of transmitter 1 and last α beams oftransmitter 2, 3 have to be aligned at receiver 1, which is shown asbelow equation.span(H ^([32]) V _([1, . . . ,M/2−α]) ^([2]))=span(H ^([31]) V_([1, . . . ,M/2−α]) ^([1]))span(H ^([23]) V _([1, . . . ,M/2−α]) ^([3]))=span(H ^([21]) V_([1, . . . ,M/2−α]) ^([1]))span(H ^([13]) V _([M/2−α+1, . . . ,M/2]) ^([3]))=span(H ^([12]) V_([M/2−α+1, . . . ,M/2]) ^([2]))  [equation 53]

Differently equation 50, equations included in the equation 53 are notcoupled with each other. If transmitter 1 uses a predetermined precoderthat is known to all receivers, first M/2−α beams at transmitter 2, 3can be easily determined as below equation.V _([1, . . . ,M/2−α]) ^([2])

span((H ^([32]))⁻¹ H ^([31]) V _([1, . . . ,M/2−α]) ^([1]))V _([1, . . . ,M/2−α]) ^([3])

span((H ^([23]))⁻¹ H ^([21]) V _([1, . . . ,M/2−α]) ^([1]))  [equation54]

In equation 54, V_([1, . . . , M/2−α]) ^([1]) predetermined precoder atnode 1 which is known at all receivers. Similarly, using predeterminedvectors for last a beams at transmitter 2, last α beams at transmitter 3can be determined as below equation.V _([M/2−α+1, . . . ,M/2]) ^([2])

span((H ^([13]))⁻¹ H ^([12]) W _([M/2−α+1, . . . ,M/2])^([2]))  [equation 55]

In equation 55, W_([M/2−α+1, . . . , M/2]) ^([2]) is a (M×α)predetermined matrix at node 2 which can be known to all receivers.Then, final precoder can be determined by using QR decomposition sincebeamforming vectors at transmitter 2,3 are satisfied with orthogonalproperty, which is shown below equation.

$\begin{matrix}{{\left\lbrack {Q^{\lbrack 2\rbrack},R} \right\rbrack = {{{{QRdecomposition}\mspace{14mu}\left\lbrack {{\left( H^{\lbrack 32\rbrack} \right)^{- 1}H^{\lbrack 31\rbrack}V_{\lbrack{1,\ldots\;,{{M/2} - \alpha}}\rbrack}^{\lbrack 1\rbrack}},W_{\lbrack{{{M/2} - \alpha + 1},\ldots\;,{M/2}}\rbrack}^{\lbrack 2\rbrack}} \right\rbrack}\left\lbrack {Q^{\lbrack 3\rbrack},R} \right\rbrack} = {{QRdecomposition}\mspace{14mu}\left\lbrack {{\left( H^{\lbrack 23\rbrack} \right)^{- 1}H^{\lbrack 21\rbrack}V_{\lbrack{1,\ldots\;,{{M/2} - \alpha}}\rbrack}^{\lbrack 1\rbrack}},{\left( H^{\lbrack 13\rbrack} \right)^{- 1}H^{\lbrack 12\rbrack}W_{\lbrack{{{M/2} - \alpha + 1},\ldots\;,{M/2}}\rbrack}^{\lbrack 2\rbrack}}} \right\rbrack}}}\mspace{20mu}{{V^{\lbrack 2\rbrack} = Q_{\lbrack{1,\ldots\;,{M/2}}\rbrack}^{\lbrack 2\rbrack}},{V^{\lbrack 3\rbrack} = Q_{\lbrack{1,\ldots\;,{M/2}}\rbrack}^{\lbrack 3\rbrack}}}} & \left\lbrack {{equation}\mspace{14mu} 56} \right\rbrack\end{matrix}$

FIG. 7 illustrates feedback information at each receiver for efficientinterference alignment method for 3 cell MIMO interference channel withlimited feedback.

Referring FIG. 7, Each transmitter transmits pilot signal and eachreceiver estimates each channel from transmitter. Based on equation 54and 55 each receiver feeds back its feedback information to targettransmitter. For example, receiver 1 feeds back(H^([13]))⁻¹H^([12])W_([M/2−α+1, . . . , M/2]) ^([2]) to transmitter 3.receiver 2 feeds back (H^([23]))⁻¹H^([21])V_([1, . . . , M/2−α]) ^([1]).And receiver 3 feeds back (H^([32]))⁻¹H^([31])V_([1, . . . , M/2−α])^([1]). In FIG. 7, FB^([i→j]) denotes feedback information from receiveri to transmitter j. By using equation 56, transmitter 2 and 3 determineeach precoder.

Efficient interference alignment method for 3 cell MIMO interferencechannel with limited feedback (hereinafter ‘proposed method’) isinvented to avoid the product of channel matrices and get betterperformance. If one node does not use α DoF, IA solution is separated toseveral independent equation, resulting in avoiding the product of allcross link channel matrices and reducing total feedback overhead.

Table. 1 shows the comparison of feedback amount between conventionalmethod (quantized channel matrix feedback) and proposed method. Theproposed method has much small feedback amount compared withconventional method.

TABLE 1 feedback amount conventional method proposed (# of feedback(channel matrix feedback) method coefficient) 6M² 2M(M/2 − α) + Mα M = 496 12 M = 8 384 48

FIG. 8 shows sum rate performance, total number of stream of othermethod.

Referring FIG. 8, the proposed method has remarkable sum rateperformance gain compared with other methods. In this result,conventional method denotes naive channel matrix feedback and CSIexchange method is the method mentioned in previous description. Totalnumber of data streams for proposed method is 5 when M=4, others is 6.For this reason, the proposed method is not only more error robust thanconventional IA solution but also reduces feedback overhead.

In previous description of section VI, it is assumed that K transmittersand K receivers are equipped with M antennas. However, hereinafter, itis considered that the K transmitters are equipped with M antennas andthe K receivers are equipped with N antennas, and the case of thehomogeneous network. Here, M and N can be an equal number or can bedifferent numbers.

Hereinafter, it is described the case of 3 user MIMO interferencechannel which is comprised of 3 transmitters and 3 receivers. The reasonwhy we consider only 3 user MIMO interference channel is that the numberof dominant interferences is not large in general cellular networks. Itis also assumed that there is no symbol extension for the simplicity intime or frequency domain. Then, node i transmits d_(i)≦min(M,N)independent spatial streams to its corresponding receiver. It is shownthat min(M,N)K DoF can be achieved if K≦R and (MN/(M+N))K DoF can beachieved if K>R where K is the number of users andR=└(max(M,N))/(min(M,N))┘

in prior arts. Without symbol extension, in pure constant MIMO channel,only an integer DoF can be achieved, therefore the optimal integerDoF(D*) is given by below equation.

$\begin{matrix}{{D^{*} = {{3{\min\left( {M,N} \right)}\mspace{14mu}{when}\mspace{14mu} R} \geq 3}}{\left\lfloor \frac{3{MN}}{M + N} \right\rfloor\mspace{14mu}{{otherwise}.}}} & \left\lbrack {{equation}\mspace{14mu} 57} \right\rbrack\end{matrix}$

The actual DoF(D) can be expressed such asD=Σ _(i=1) ³ d _(i)

where d_(i) is the number of spatially independent data streams attransmitter i. The actual DoF must be less and equal to D*. Under ourassumptions, the received signal at receiver k can be written as belowequation.

$\begin{matrix}{y_{i} = {{\sqrt{\frac{P}{d_{i\;}}}H_{i,i}V_{i}s_{i}} + {E_{l \neq i}\sqrt{\frac{P}{d_{l}}}H_{i,l}V_{l}s_{l}} + {n_{i}.}}} & \left\lbrack {{equation}\mspace{14mu} 58} \right\rbrack\end{matrix}$

In equation 58, y_(i) is the N×1 received signal vector, P is transmitpower, H_(i,j) is the N×M channel matrix from transmitter 1 to receiveri, V_(i) is the M×d_(i) precoding matrix used at transmitter i, s_(i) isthe d_(i)×1 transmitted symbol vector at transmitter i, with unit normelement, i.e,

(s _(i) ^(H) s _(i))=d _(i),

and n_(i) is a complex Gaussian noise vector at receiver i withcovariance matrix σ²I_(N). Here, the notation H_(m,n) has same meaningwith the previous notation H^([mn]). And the notation V_(i) is same tothe previous notation V^([i]).

The suboptimal sum rate due to imperfect CSIT (channel statusinformation at transmitter) achieved by the linear zero-forcing receiveru, which is given by below equation.

$\begin{matrix}{R_{sum} = {\sum\limits_{i = 1}^{3}{\sum\limits_{m = 1}^{d_{i}}{\log_{2}\left( {1 + \frac{\frac{P}{d_{i}}{{\left( u_{i}^{m} \right)^{H}H_{i,i}v_{i}^{m}}}^{2}}{{\sum\limits_{l \neq m}{\frac{P}{d_{i}}{{\left( u_{i}^{m} \right)^{H}H_{i,i}v_{i}^{l}}}^{2}}} + {\sum\limits_{k \neq i}{\sum\limits_{l = 1}^{d_{k}}{\frac{P}{d_{k}}{{\left( u_{i}^{m} \right)^{H}H_{i,k}v_{k}^{l}}}^{2}}}} + \sigma^{2}}} \right)}}}} & \left\lbrack {{equation}\mspace{14mu} 59} \right\rbrack\end{matrix}$

If CSIT is perfect, first and second term of denominator of logarithmfunction in equation 59 will be zero.

Hereinafter, a set of interference coordination methods are describedfor different transmit/receive antenna configurations.

A. Case 1: M≦N and D*≦N

For this case, there are extra DoF at each receiver such that achievingD* does not require IA (interference alignment). Also there is no needof CSIT for achieving D*. Although using any random precoders attransmitter, all operation to cancel interferences can be done atreceiver side.

B. Case 2: M≦N and D*>N

If there exists achievable DoF vector d=[d₁,d₂,d₃] to satisfy one amongthe following two constraints, then the interference coordination methodexplained below can achieve the optimal integer DoF(D*) and reduction ofCSI error amplication.[contraints 1]2d ₁+3d ₂≦2N when d ₁ ≦d ₂ and d ₃ =d ₂  1.3d ₁+2d ₃≦2N when d ₁=d₂ and d ₁ ≦d ₃.  2.

Under either of the above two constraints, D* can be achieved by thefollowing interference coordination method based on interferencealignment. Above constraints can be derived intuitively. Let considerfollowing parameter,λ=d ₁ +d ₂ +d ₃ −N,

where it is the number of overlap dimension means that at least λ beamsshould be aligned at each receiver. CSI error amplification issue comesfrom coupling condition of IA. If λ=d_(i), all beams at each transmitterare coupled with other beams from other 2 transmitters, which results inthe requirement of all cross link CSIT at all transmitters. However, if2λ≦d₂, i.e. 2d₁+3d₂≦2N, when d₁≦d₂, d₃=d₂, it is sufficient that eachbeam is aligned with at most one beam of other transmitter. First λbeams at transmitter 2, 3 are aligned with that of transmitter 1, andrest λ beams at transmitter 2 are aligned with that of transmitter 3. Attransmitter 2, first λ beams and rest λ beams should be disjoint toavoid multiple coupling of each beam, therefore 2λ≦d₂ should besatisfied. Under this constraint, to determine each precoder at eachtransmitters, needs only CSI from one receiver. Second constraint3d₁+2d₃≦2N can be obtain by similar way. For instance D*=5 when M=3, N=4case, d[1,2,2] is satisfied with first constraint such configuration canbe achievable D*. If some antenna configuration can not be satisfiedwith above constraint, reduce actual DoF D until satisfy the constraint,leads to D=2N−d₁−d₂<D* to be achievable, such configuration can notachieve D* but reduce error variance. Based on this, we proposeinterference coordination method 1 comprising below steps.

1. step 1 (Set up): Each receiver checks existence of achievable DoFvector d=[d₁,d₂,d₃] satisfied with one of the constraints (i.e.2d₁+3d₂≦2N, d₁≦d₂, d₃=d₂ or 3d₁+2d₃≦2N, d₁=d₂, d₁≦d₃). If there existsd, D=D* is achievable. Else each receiver reduces D until satisfied withone of the constraints.

2. step 2: For aεA, A=[a₁, . . . . , a_(λ)|∀a_(i)ε[1, . . . , d₂],a₁≠a₂≠ . . . ≠a_(λ)], computation of v₂ ^(a) at receiver 3 to be alignedwith v₁ ^(a), which is predetermined vector already known to receiver 2,3 and computation of v₃ ^(a) at receiver 2 to be aligned with v₁ ^(a).For bεB,B=[b ₁ , . . . ,b _(λ) |∀b _(i)ε[1, . . . ,d ₂ ]\A,b ₁ ≠b ₂ ≠ . . . ≠b_(λ)],whereA\Bdenotes a set from A excluding B, computation of v₃ ^(b) at receiver 1to be aligned with v₂ ^(b) which is predetermined vector already knownto receiver 1.

3. Step 3: Feedback v₂ ^(a), v₃ ^(a) and v₃ ^(b) vectors tocorresponding transmitters.

In full channel feedback method, each receiver feeds back 2MN non zerocoefficient to 3 transmitters. Since there are 3 receivers, totalfeedback overhead is 18MN. However, in the above method receiver feedsback λM non zeros coefficient to its corresponding transmitter. Totalfeedback overhead for the proposed algorithm is 3λM. Also there is muchreduction of error propagation issue on the proposed method sincedetermining each beam requires the CSIT from only 1 receiver.

C. Case 3: M>N and D*<M

For this case, there are extra DoF at each transmitter such thatachieving D*. Differently case 1, this case require CSIT. By usingreciprocity, feedback overhead can be efficiently reduced. We proposeinterference coordination method 2 comprising below steps.

1. Step 1: Each receiver determines receiver beamformer U_(i) firstramdomly or maximizing desired channel SINR.

2. Step 2: Each receiver feeds back U_(i) ^(H)H_(i,k), k≠i tocorresponding transmitters.

3. Step 3:

${v_{i}^{m} = {v_{m}\left\lbrack {\overset{3}{\sum\limits_{{k = 1},{k \neq i}}}{\frac{P}{d^{i}}H_{k,i}^{H}U_{k}U_{k}^{H}H_{k,i}}} \right\rbrack}},{m = 1},{\ldots\mspace{14mu} d_{i}}$where v_(m)[A] is the eigenvector corresponding to the d^(th) smallesteigenvalue of A. In the proposed method each receiver feeds back d_(i)Mnon-zero coefficient to 2 transmitters. Total feedback overhead for theproposed algorithm is 6D*M.

D. Case 4: M>N and D*≧M

In this case, each precoder need be determined for interferencecancellation and alignment suitably. Receive beamformer can be moreefficiently determined than transmit beamformer since consideringreciprocal channel receiver side consider only interference alignmentand size of receive beamformer is less than that of transmit beamformer.Although considering reciprocal channel, there is issue of erroramplification. To avoid error amplification, each receiver can checkfollowing constraints.[constraints 2]2d ₁+3d ₂≦2M,d ₁ ≦d ₂ ,d ₃ =d ₂  13d ₁+2d ₃≦2M,d ₁ =d ₂ ,d ₁ ≦d ₃.  2

Differently case 2, N changes to M because of reciprocal channelproperty, also λ=d₁+d₂+d₃−M=D−M. After setup with satisfied with aboveconstraints, receive beamformers are firstly determined, then eachreceiver feeds back its effective channel to corresponding transmitterslike case 3. We propose interference coordination method 3 comprisingbelow steps.

1. step 1(Set up): Each receiver check existence of achievable DoFvector d=[d₁,d₂,d₃] satisfied with constraints 2(2d₁+3d₂≦2M, d₁≦d₂,d₃=d₂ or 3d₁+2d₃≦2M, d₁=d₂, d₁≦d₃). If there exists d, D=D* isachievable, else reduce D until satisfied with one of the constraints 2.

2. Step 2: ForaεA,A=[a ₁ , . . . ,a _(λ) |∀a _(i)ε[1, . . . ,d ₂ ],a ₁ ≠a ₂ ≠ . . . ≠a_(λ)],

receiver 1 feeds back (U₁ ^(a))^(H) H_(1,3) to receiver 2 forcomputation of u₂ ^(a) to be aligned with u₁ ^(a) at transmitter 3,receiver 1 feeds back (U₁ ^(a))^(H) H_(1,2) for computation of u₃ ^(a)to be aligned with u₁ ^(a) at transmitter 2, where u₁ ^(a) ispredetermined.

ForbεB,B=[b ₁ , . . . ,b _(λ) |∀b _(i)ε[1, . . . ,d ₂ ]\A,b ₁ ≠b ₂ ≠ . . .≠b _(λ)],

receiver 2 feeds back (u₂ ^(b))^(H)H_(2,1) for computation of u₃ ^(b) tobe aligned with u₂ ^(b) at transmitter 1, where u₂ ^(b) ispredetermined.

3. Step 3: Each receiver feeds back U_(i) ^(H)H_(i,k), k≠i tocorresponding transmitters.

4. Step 4:

$v_{i}^{m} = {v_{m}\left\lbrack {\sum\limits_{{k = 1},{k \neq i}}^{3}{\frac{P}{d^{i}}H_{k,i}^{H}U_{k}U_{k}^{H}H_{k,i}}} \right\rbrack}$m=1, . . . d_(i), where v_(m)[A] is the eigenvector corresponding to thed^(th) smallest eigenvalue of A.

In step 3, feedback overhead is 3λM, then step 4 feedback overhead isΣ_(i=1) ³2d _(i) M=2DM.

Total feedback overhead for the proposed algorithm is 3λM+2DM=5DM−3M².For uplink system, if wireline backhaul between each base station isassumed so that each receiver shares each channel information, feedbackoverhead in step 3 can be reduced to zero.

While the invention has been described in connection with what ispresently considered to be practical exemplary embodiments, it is to beunderstood that the invention is not limited to the disclosedembodiments, but, on the contrary, is intended to cover variousmodifications and equivalent arrangements included within the spirit andscope of the appended claims.

The invention claimed is:
 1. A method for interference alignment inwireless network having 3 transmitters and 3 receivers which areequipped with M antennas, the method comprising: transmitting, performedby each of the 3 transmitters, a pilot signal known to the 3 receivers;estimating, performed by each of the 3 receivers, each channel fromtransmitter; transmitting, performed by each of the 3 receivers,feedback information to target transmitter; and determining, performedby transmitter 2 and transmitter 3, a precoding vector; wherein a degreeof freedom (DoF) of a transmitter 1 is (M/2−α), a DoF of the transmitter2 or the transmitter 3 is M/2.
 2. The method of claim 1, wherein first(M/2−α) beams at the transmitter 2 and the transmitter 3 are alignedwith beams of the transmitter
 1. 3. The method of claim 1, wherein lasta beams of the transmitter 2 and the transmitter 3 are aligned at areceiver
 1. 4. The method of claim 1, wherein a transmitter 1 uses apredetermined precoder which is known to the 3 receivers.
 5. The methodof claim 4, wherein the predetermined precoder is (M×(M/2−α)) matrix. 6.The method of claim 5, wherein the feedback information comprisesinformation expressed below equation: [equation]FB^([1→3])=(H ^([13]))⁻¹ H ^([12]) W _([M/2−α+1, . . . ,M/2]) ^([2])FB^([2→3])=(H ^([23]))⁻¹ H ^([21]) V _([1, . . . ,M/2−α]) ^([1])FB^([3→2])=(H ^([32]))⁻¹ H ^([31]) V _([1, . . . ,M/2−α]) ^([1]) whereFB^([i→j]) denotes feedback information from receiver i to transmitterj, W_([M/2−α+1, . . . , M/2]) ^([2]) is (M×α) predetermined matrix atthe transmitter 2, H^([mn]) is a channel matrix between transmitter nand receiver m and V_([1, . . . , M/2−α]) ^([1]) is the (M×(M/2−α))predetermined precoder of the transmitter
 1. 7. The method of claim 6,wherein first (M/2−α) beams at the transmitter 2 and the transmitter 3is determined by below equation: [equation]V _([1, . . . ,M/2−α]) ^([2])

span((H ^([32]))⁻¹ H ^([31]) V _([1, . . . ,M/2−α]) ^([1]))V _([1, . . . ,M/2−α]) ^([3])

span((H ^([23]))⁻¹ H ^([21]) V _([1, . . . ,M/2−α]) ^([1])).
 8. Themethod of claim 7, wherein last α beams at the transmitter 3 isdetermined by below equation: [equation]V _([M/2−α+1, . . . ,M/2]) ^([3])

span((H ^([13]))⁻¹ H ^([12]) W _([M/2−α+1, . . . ,M/2]) ^([2])).
 9. Themethod of claim 8, wherein the precoder of the transmitter 2 and theprecoder of the transmitter 3 are determined by the below equation:[equation][Q^([2]), R] = QRdecomposition[(H^([32]))⁻¹H^([31])V_([1, …  , M/2 − α])^([1]), W_([M/2 − α + 1, …  , M/2])^([2])][Q^([3]), R] = QRdecompositon[(H^([23]))⁻¹H^([21])V_([1, …  , M/2 − α])^([1]), (H^([13]))⁻¹H^([12])W_([M/2 − α + 1, …  , M/2])^([2])]V^([2]) = Q_([1, …  , M/2])^([2]), V^([3]) = Q_([1, …  , M/2])^([3]).